Camera View Coordinates and Z-depth values
The Local Axis for the camera is as you describe: Negative Z is the direction the camera is looking at (the camera local axis is independent of the global axis, and might not be aligned to the global axis at all).

If you move the camera using positive values on it's local Z axis , the camera moves back (away from wherever it is pointing at)



The local Z Axis for the camera and and Z-Depth on the image created by the camera are no the same "Z".

The Z-Depth values (or  Z-pass) are always in positive numbers, they represents the absolute distance from the camera to geometry in the scene for every pixel of the image. The camera is the 0 on the scale and the units used for its values are the working units set for the scene (blender units, meters, cm, inches, etc) The minimum and maximum value are clipped according to the camera's start and end clip distances.

Z Depth does not give you any information on the coordinates of the objects on the scene nor does it give you any information on the coordinates or rotation of the camera.

Just to clarify... Z-Depth values are positive values away from the camera (in its negative local Z axis...) and all of these Zs are completely independent of the World's Z axis.